// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2014 Pedro Gonnet (pedro.gonnet@gmail.com)
// Copyright (C) 2016 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_MATHFUNCTIONSIMPL_H
#define EIGEN_MATHFUNCTIONSIMPL_H

namespace Eigen {

namespace internal {

/** \internal \returns the hyperbolic tan of \a a (coeff-wise)
	Doesn't do anything fancy, just a 13/6-degree rational interpolant which
	is accurate up to a couple of ulps in the (approximate) range [-8, 8],
	outside of which tanh(x) = +/-1 in single precision. The input is clamped
	to the range [-c, c]. The value c is chosen as the smallest value where
	the approximation evaluates to exactly 1. In the reange [-0.0004, 0.0004]
	the approxmation tanh(x) ~= x is used for better accuracy as x tends to zero.

	This implementation works on both scalars and packets.
*/
template<typename T>
T
generic_fast_tanh_float(const T& a_x)
{
	// Clamp the inputs to the range [-c, c]
#ifdef EIGEN_VECTORIZE_FMA
	const T plus_clamp = pset1<T>(7.99881172180175781f);
	const T minus_clamp = pset1<T>(-7.99881172180175781f);
#else
	const T plus_clamp = pset1<T>(7.90531110763549805f);
	const T minus_clamp = pset1<T>(-7.90531110763549805f);
#endif
	const T tiny = pset1<T>(0.0004f);
	const T x = pmax(pmin(a_x, plus_clamp), minus_clamp);
	const T tiny_mask = pcmp_lt(pabs(a_x), tiny);
	// The monomial coefficients of the numerator polynomial (odd).
	const T alpha_1 = pset1<T>(4.89352455891786e-03f);
	const T alpha_3 = pset1<T>(6.37261928875436e-04f);
	const T alpha_5 = pset1<T>(1.48572235717979e-05f);
	const T alpha_7 = pset1<T>(5.12229709037114e-08f);
	const T alpha_9 = pset1<T>(-8.60467152213735e-11f);
	const T alpha_11 = pset1<T>(2.00018790482477e-13f);
	const T alpha_13 = pset1<T>(-2.76076847742355e-16f);

	// The monomial coefficients of the denominator polynomial (even).
	const T beta_0 = pset1<T>(4.89352518554385e-03f);
	const T beta_2 = pset1<T>(2.26843463243900e-03f);
	const T beta_4 = pset1<T>(1.18534705686654e-04f);
	const T beta_6 = pset1<T>(1.19825839466702e-06f);

	// Since the polynomials are odd/even, we need x^2.
	const T x2 = pmul(x, x);

	// Evaluate the numerator polynomial p.
	T p = pmadd(x2, alpha_13, alpha_11);
	p = pmadd(x2, p, alpha_9);
	p = pmadd(x2, p, alpha_7);
	p = pmadd(x2, p, alpha_5);
	p = pmadd(x2, p, alpha_3);
	p = pmadd(x2, p, alpha_1);
	p = pmul(x, p);

	// Evaluate the denominator polynomial q.
	T q = pmadd(x2, beta_6, beta_4);
	q = pmadd(x2, q, beta_2);
	q = pmadd(x2, q, beta_0);

	// Divide the numerator by the denominator.
	return pselect(tiny_mask, x, pdiv(p, q));
}

template<typename RealScalar>
EIGEN_DEVICE_FUNC EIGEN_STRONG_INLINE RealScalar
positive_real_hypot(const RealScalar& x, const RealScalar& y)
{
	// IEEE IEC 6059 special cases.
	if ((numext::isinf)(x) || (numext::isinf)(y))
		return NumTraits<RealScalar>::infinity();
	if ((numext::isnan)(x) || (numext::isnan)(y))
		return NumTraits<RealScalar>::quiet_NaN();

	EIGEN_USING_STD(sqrt);
	RealScalar p, qp;
	p = numext::maxi(x, y);
	if (p == RealScalar(0))
		return RealScalar(0);
	qp = numext::mini(y, x) / p;
	return p * sqrt(RealScalar(1) + qp * qp);
}

template<typename Scalar>
struct hypot_impl
{
	typedef typename NumTraits<Scalar>::Real RealScalar;
	static EIGEN_DEVICE_FUNC inline RealScalar run(const Scalar& x, const Scalar& y)
	{
		EIGEN_USING_STD(abs);
		return positive_real_hypot<RealScalar>(abs(x), abs(y));
	}
};

// Generic complex sqrt implementation that correctly handles corner cases
// according to https://en.cppreference.com/w/cpp/numeric/complex/sqrt
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T>
complex_sqrt(const std::complex<T>& z)
{
	// Computes the principal sqrt of the input.
	//
	// For a complex square root of the number x + i*y. We want to find real
	// numbers u and v such that
	//    (u + i*v)^2 = x + i*y  <=>
	//    u^2 - v^2 + i*2*u*v = x + i*v.
	// By equating the real and imaginary parts we get:
	//    u^2 - v^2 = x
	//    2*u*v = y.
	//
	// For x >= 0, this has the numerically stable solution
	//    u = sqrt(0.5 * (x + sqrt(x^2 + y^2)))
	//    v = y / (2 * u)
	// and for x < 0,
	//    v = sign(y) * sqrt(0.5 * (-x + sqrt(x^2 + y^2)))
	//    u = y / (2 * v)
	//
	// Letting w = sqrt(0.5 * (|x| + |z|)),
	//   if x == 0: u = w, v = sign(y) * w
	//   if x > 0:  u = w, v = y / (2 * w)
	//   if x < 0:  u = |y| / (2 * w), v = sign(y) * w

	const T x = numext::real(z);
	const T y = numext::imag(z);
	const T zero = T(0);
	const T w = numext::sqrt(T(0.5) * (numext::abs(x) + numext::hypot(x, y)));

	return (numext::isinf)(y) ? std::complex<T>(NumTraits<T>::infinity(), y)
		   : x == zero		  ? std::complex<T>(w, y < zero ? -w : w)
		   : x > zero		  ? std::complex<T>(w, y / (2 * w))
							  : std::complex<T>(numext::abs(y) / (2 * w), y < zero ? -w : w);
}

// Generic complex rsqrt implementation.
template<typename T>
EIGEN_DEVICE_FUNC std::complex<T>
complex_rsqrt(const std::complex<T>& z)
{
	// Computes the principal reciprocal sqrt of the input.
	//
	// For a complex reciprocal square root of the number z = x + i*y. We want to
	// find real numbers u and v such that
	//    (u + i*v)^2 = 1 / (x + i*y)  <=>
	//    u^2 - v^2 + i*2*u*v = x/|z|^2 - i*v/|z|^2.
	// By equating the real and imaginary parts we get:
	//    u^2 - v^2 = x/|z|^2
	//    2*u*v = y/|z|^2.
	//
	// For x >= 0, this has the numerically stable solution
	//    u = sqrt(0.5 * (x + |z|)) / |z|
	//    v = -y / (2 * u * |z|)
	// and for x < 0,
	//    v = -sign(y) * sqrt(0.5 * (-x + |z|)) / |z|
	//    u = -y / (2 * v * |z|)
	//
	// Letting w = sqrt(0.5 * (|x| + |z|)),
	//   if x == 0: u = w / |z|, v = -sign(y) * w / |z|
	//   if x > 0:  u = w / |z|, v = -y / (2 * w * |z|)
	//   if x < 0:  u = |y| / (2 * w * |z|), v = -sign(y) * w / |z|

	const T x = numext::real(z);
	const T y = numext::imag(z);
	const T zero = T(0);

	const T abs_z = numext::hypot(x, y);
	const T w = numext::sqrt(T(0.5) * (numext::abs(x) + abs_z));
	const T woz = w / abs_z;
	// Corner cases consistent with 1/sqrt(z) on gcc/clang.
	return abs_z == zero ? std::complex<T>(NumTraits<T>::infinity(), NumTraits<T>::quiet_NaN())
		   : ((numext::isinf)(x) || (numext::isinf)(y)) ? std::complex<T>(zero, zero)
		   : x == zero									? std::complex<T>(woz, y < zero ? woz : -woz)
		   : x > zero									? std::complex<T>(woz, -y / (2 * w * abs_z))
					  : std::complex<T>(numext::abs(y) / (2 * w * abs_z), y < zero ? woz : -woz);
}

template<typename T>
EIGEN_DEVICE_FUNC std::complex<T>
complex_log(const std::complex<T>& z)
{
	// Computes complex log.
	T a = numext::abs(z);
	EIGEN_USING_STD(atan2);
	T b = atan2(z.imag(), z.real());
	return std::complex<T>(numext::log(a), b);
}

} // end namespace internal

} // end namespace Eigen

#endif // EIGEN_MATHFUNCTIONSIMPL_H
